Cyclic orbifolds of lattice vertex operator algebras having group like fusions

TL;DR

This paper characterizes when orbifold vertex operator algebras derived from lattice VOAs have group-like fusion, determines their fusion rings, and applies these results to specific lattices including the Leech lattice, confirming a conjecture.

Contribution

It provides a complete criterion for group-like fusion in orbifold VOAs of lattice type and explicitly describes their fusion structures, advancing understanding of their algebraic properties.

Findings

Orbifold VOAs have group-like fusion iff the automorphism acts trivially on the discriminant group.

Fusion rings and quadratic space structures are explicitly determined for fixed point free automorphisms.

The method confirms a conjecture related to coinvariant sublattices of the Leech lattice.

Abstract

Let $L$ be an even (positive definite) lattice and $g\in O(L)$. In this article, we prove that the orbifold vertex operator algebra $V_{L}^{\hat{g}}$ has group-like fusion if and only if $g$ acts trivially on the discriminant group $\mathcal{D}(L)=L^*/L$ (or equivalently $(1-g)L^*<L$). We also determine their fusion rings and the corresponding quadratic space structures when $g$ is fixed point free on $L$. By applying our method to some coinvariant sublattices of the Leech lattice $\Lambda$, we prove a conjecture proposed by G. H\"ohn. In addition, we also discuss a construction of certain holomorphic vertex operator algebras of central charge $24$ using the the orbifold vertex operator algebra $V_{\Lambda_g}^{\hat{g}}$.